Proceedings of the Edinburgh Mathematical Society最新文献

英文中文

Solid bases and functorial constructions for (p-)Banach spaces of analytic functions 解析函数 (p-)Banach 空间的实体基和函数构造

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-09-09 DOI: 10.1017/s001309152400035x

Guozheng Cheng, Xiang Fang, Chao Liu, Yufeng Lu

Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials

${z^n}_{ngeq 0}$

exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of

$mathcal{X}^mathrm{max}$

(and

$mathcal{X}^mathrm{min}$

), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.

单位盘上的巴拿赫函数空间是随机解析函数研究中的符号空间，其单项式 ${z^n}_{ngeq 0}$ 表现出无条件基的特征，但它们通常甚至不构成一个肖德基，受这些新例子的启发，我们为巴拿赫空间和 p-Banach 空间引入了一个称为实基的概念，并研究了它的性质。除了证明固态基的丰富存在性之外，我们还研究了固态基与无条件基的关系、泰勒多项式的弱星收敛、固态跨度问题以及 c0 扮演的奇特角色。这项工作有以下两个特点：(1) 在这一过程中，我们重新审视了单位盘上解析函数的典型巴拿赫空间所满足的公理，从而得出了 $mathcal{X}^mathrm{max}$ （和 $mathcal{X}^mathrm{min}$ ）的概念，以及一些相关的函数构造，这些都是我们感兴趣的；(2) 坚实基础的主要意义在于不可分离的（p-）巴拿赫空间，例如 BMOA 和布洛赫空间，而不是 VMOA 和小布洛赫空间。

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引用次数: 0

Equisingularity in pencils of curves on germs of reduced complex surfaces 还原复曲面胚芽上曲线铅笔的等差数列

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-06-04 DOI: 10.1017/s0013091524000245

Gonzalo Barranco Mendoza, Jawad Snoussi

We study pencils of curves on a germ of complex reduced surface $(S,0)$. These are families of curves parametrized by $ mathbb{P}^1 $ having 0 as the unique common point. We prove that for $win mathbb{P}^1$, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$, where f and g are generators of the pencil.

我们研究复还原曲面$(S,0)$胚芽上的曲线铅笔。这些曲线是以mathbb{P}^1 $ 为参数、以 0 为唯一公共点的曲线族。我们证明，对于 $win mathbb{P}^1$，铅笔的相应曲线不具有泛函拓扑，当且仅当拉回铅笔到归一化曲面的相应曲线具有非泛函拓扑，或者 w 是函数 $ f/g $ 沿 $(S,0)$ 的奇点位置的极限值（其中 f 和 g 是铅笔的生成器）。

引用次数: 0

A classification of automorphic Lie algebras on complex tori 复杂环上的非定常李代数分类

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-28 DOI: 10.1017/s0013091524000324

Vincent Knibbeler, Sara Lombardo, Casper Oelen

We classify the automorphic Lie algebras of equivariant maps from a complex torus to

$mathfrak{sl}_2(mathbb{C})$

. For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of

$mathrm{PSL}_2({mathbb{Z}})$

, apart from four cases, which are all isomorphic to Onsager’s algebra.

我们对从复环面到 $mathfrak{sl}_2(mathbb{C})$ 的等变映射的自形李代数进行了分类。对于每种情况，我们都计算出一个正则表达式的基。除了与昂萨格代数同构的四种情况之外，这些自变分李代数精确地对应于以 $mathrm{PSL}_2({mathbb{Z}}) $ 的模态曲线为参数的两个不相邻的李代数族。

引用次数: 0

Coactions and skew products for topological quivers 拓扑四元组的共生和倾斜积

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-22 DOI: 10.1017/s0013091524000208

Lucas Hall

Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the

${C^*}$

-algebra of the skew product and a certain native coaction on the

${C^*}$

-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the

${C^*}$

-algebra of the original quiver.

给定局部紧密群在拓扑簇上的循环，作者构建了一个斜积拓扑簇，并确定了拓扑簇可以被识别为斜积的条件。我们研究了斜积的 ${C^*}$ - 代数与原簇的 ${C^*}$ - 代数上的某个本征协效之间的关系，发现协效的交叉积与斜积同构。作为应用，我们证明了对偶作用的还原交叉积与原始簇的 ${C^*}$ - 代数是莫里塔等价的。

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引用次数: 0

Characterization of continuous homomorphisms on entire slice monogenic functions 整片单原函数上连续同态的特征

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-17 DOI: 10.1017/s0013091524000373

Stefano Pinton, Peter Schlosser

This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

本文的灵感来自量子力学中出现的一类无穷阶微分算子。在研究量子场方程的超振荡演化时，它们被证明是一种重要工具。无穷阶微分算子自然地作用于全函数空间或超函数空间。最近，在全单元函数空间（即狄拉克算子内核中的函数）上考虑和描述了无穷阶微分算子。本文的重点是描述连续作用于另一类超全同形函数的无穷阶微分算子的特征，这类超全同形函数被称为在克利福德代数中具有值的切片超全同形函数或切片单元函数。这种函数理论具有非常广泛的相关谱理论，在这种情况下，函数理论和算子理论都受到了深入研究。在这里，我们引入了近似阶的概念，并建立了全片单元函数的一些基本性质，这些性质对于作用于全片单元函数的无穷阶微分算子的表征至关重要。

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引用次数: 0

On the smoothness of slowly varying functions 论缓慢变化函数的平稳性

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-16 DOI: 10.1017/s0013091524000348

Dalimil Peša

In this paper, we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.

在本文中，我们考虑了符合现代定义的缓变函数的平滑性问题，在过去二十年中，该定义在函数空间和插值的应用中得到了普及。我们证明，每一个这种类型的慢变函数都等价于一个具有所有阶连续经典导数的慢变函数。

引用次数: 0

On the convexity of the quaternionic essential numerical range 论四元基本数值范围的凸性

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-15 DOI: 10.1017/s0013091524000336

LuÍs Carvalho, Cristina Diogo, Sérgio Mendes, Helena Soares

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

一般来说，四元数环境中的数值范围是四元数的一个非凸子集。本质数值范围是数值范围的细化，只保留在某种意义上具有无限倍性的元素。我们证明了四元数希尔伯特空间上有界线性算子的基本数值范围是凸的。我们还提供了兰卡斯特定理的四元数类似定理，它涉及数值范围的闭合及其基本数值范围。

引用次数: 0

Powers of commutators in linear algebraic groups 线性代数群中换元的幂

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-14 DOI: 10.1017/s0013091524000361

Benjamin Martin

Let ${mathcal G}$ be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let $G= {mathcal G}(k)$. We prove that if $gammain G$ such that γ is a commutator and $deltain G$ such that $langle deltarangle= langle gammarangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.

让 ${mathcal G}$ 是一个 k 上的线性代数群，其中 k 是一个代数闭域、伪无限域或非拱顶局部域的估值环。让 $G= {mathcal G}(k)$.我们证明，如果 $gammain G$ 使得 γ 是换元器，并且 $deltain G$ 使得 $langledeltarangle= langlegammarangle$ 那么 δ 是换元器。这概括了本田对有限群的一个结果。我们的证明使用了一阶模型理论中的 Lefschetz 原则。

引用次数: 0

Growth of hypercyclic functions: a continuous path between -frequent hypercyclicity and hypercyclicity 超循环函数的增长：频繁超循环与超循环之间的连续路径

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-08 DOI: 10.1017/s0013091524000312

Augustin Mouze, Vincent Munnier

We are interested in the optimal growth in terms of Lp-averages of hypercyclic and

$mathcal{U}$

-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between

$mathcal{U}$

-frequent hypercyclicity and hypercyclicity.

我们对作用于单位圆盘上解析函数空间的某些加权泰勒移位算子的超循环函数和 $mathcal{U}$ -频繁超循环函数的 Lp 平均值的最优增长感兴趣。我们通过考虑介于 $mathcal{U}$ -频繁超周期性和超周期性之间的上频繁超周期性中间概念，统一了所得到的结果。

引用次数: 0

Some congruences for 12-coloured generalized Frobenius partitions 12 色广义弗罗贝尼斯分区的一些全等式

IF 0.7 3区数学 Q2 MATHEMATICS

Proceedings of the Edinburgh Mathematical Society

Pub Date : 2024-05-02 DOI: 10.1017/s0013091524000294

Su-Ping Cui, Nancy S. S. Gu, Dazhao Tang

In his 1984 AMS Memoir, Andrews introduced the family of functions

$cphi_k(n)$

, the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of

$textrm{C}Phi_k(q)$

for

$2leq kleq17$

by utilizing the theory of modular forms, where

$textrm{C}Phi_k(q)$

denotes the generating function of

$cphi_k(n)$

. In this paper, we first establish another expression of

$textrm{C}Phi_{12}(q)$

with integer coefficients, then prove some congruences modulo small powers of 3 for

$cphi_{12}(n)$

by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by

$cphi_{12}(n)$

安德鲁斯在1984年的AMS回忆录中介绍了函数$cphi_k(n)$族，即n的k色广义弗罗贝纽斯分区数。2019年，Chan、Wang和Yang利用模形式理论系统地研究了$textrm{C}Phi_k(q)$对于$2leq kleq17$的算术性质，其中$textrm{C}Phi_k(q)$表示$cphi_k(n)$的生成函数。本文首先建立了$textrm{C}Phi_{12}(q)$ 的另一个整数系数表达式，然后利用阿拉卡（A. Alaca）、阿拉卡（S. Alaca）和威廉姆斯（Williams）的θ函数的一些参数化同调，证明了$cphi_{12}(n)$ 的一些小幂次同调。最后，我们猜想$cphi_{12}(n)$ 满足三个 3 的幂的全等族。

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引用次数: 0

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